A 250-g block fits inside a small cavity cut in arm OA, which rotates in the vertical plane at a constant rate such that ν = 3 m/s. Knowing that the spring exerts on block B a force of magnitude P =1.5 N and neglecting the effect of friction, determine the range of values of θ for which block B is

Question

A 250-g block fits inside a small cavity cut in arm OA, which rotates in the vertical plane at a constant rate such that [latex] u=3 \ m/s[/latex]. Knowing that the spring exerts on block B a force of magnitude [latex]P=1.5 \ N[/latex] and neglecting the effect of friction, determine the range of values of [latex] heta[/latex] for which block B is in contact with the face of the cavity closest to the axis of rotation O.

Accepted Answer

[latex] ext { +} \swarrow \Sigma F_{n}=m a_{n}: \quad P+m g \sin heta-Q=m \frac{ u^{2}}{ ho}[/latex]

To have contact with the specified surface, we need Q ≥ 0,

or [latex]\quad \quad Q=P+mg\;\mathrm{sin}\, heta-\frac{m u^{2}}{ ho}\gt 0[/latex]

[latex]\sin heta\gt \frac{1}{g}\left\lgroup\frac{ u^{2}}{ ho}-\frac{P}{m} ight group[/latex] (1)

Data:[latex]\quad \quad m=0.250 \mathrm{~kg}, \quad u=3 \mathrm{~m} / \mathrm{s}, \quad P=1.5 \mathrm{~N}, \quad ho=0.9 \mathrm{~m}[/latex]

Substituting into (1):

[latex]\begin{aligned}& \sin heta>\frac{1}{9.81}\left[\frac{(3)^{2}}{0.9}-\frac{1.5}{0.25} ight] \& \sin heta>0.40775\end{aligned}[/latex]

[latex]24.1^{\circ}< heta<155.9^{\circ}\blacktriangleleft[/latex]

Question 12.48: A 250-g block fits inside a small cavity cut in arm OA, whic...

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